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Are wings any more efficient at creating lift, versus orienting the engine's thrust downwards?
As I understood from this article, wings on an airplane basically create lift by pushing air down.
Maintaining the lift force expends energy because air needs to be continually accelerated downwards. In the "energy budget" of the airplane, this energy cost appears in the form of extra drag due to the wing, which must be compensated by extra engine thrust.
It seems to me then that the power required to keep an airplane at a constant altitude using a wing is exactly equivalent to the power that would be needed if we would "simply" use a downwards-pointing fan instead of the wing.
Now, some posts on this website compare VTOL aircraft to airliners of similar size, and state that VTOL requires much more thrust (e.g. here - I would post more links but I'm not allowed to). But according to the reasoning above, shouldn't it require exactly the same amount of power to put a plane in the air, whether using a wing or directing the engine thrust downwards?
Please disregard, for a moment, the technical difficulties and safety issues. They are discussed in other questions related to gimbal thrust and VTOL aircraft. My question has to do with energy expenditure and power.

Pretty darn good for a first question! Welcome!

Note that a helicopter is like a prop plane with no wings and a huge prop pointed down. So yes, you can create useful lift by just pointing an engine straight down, but then you can't use the same engine as well for forward motion. This is assuming your want your aircraft to take you somewhere besides just up. Compare with rockets used for spaceflight.

weeeelllll... @ToddWilcox, helicopters do move forward, they just angle their wings in the direction they want to go. Gimbaled engines (or exhaust) would do the same thing.

Thanks a lot for all the interest and contributions! I'll check out the answers and comment on them, but right now I think the question still lacks closure as far as the underlying physics is concerned.

@FreeMan I should have written, "you can't use the same engine as effectively for forward motion." Meaning, it's not as efficient for traveling quickly in a horizontal direction. When you use a gimbaled engine system the aircraft basically becomes fixed wing in flight.

Re "...shouldn't it require exactly the same amount of power...": Yes, and bumblebees shouldn't be able to fly :-)  Nevertheless, we observe bumblebees flying, and winged aircraft using much less power than would be needed to keep the aircraft aloft through thrust.  Even VTOL aircraft transition to using wings for lift once they're off the ground.

Technically, downward thrust does not create lift. It does of course have the same effect as lift in that it counters the force of gravity, but it's not lift :)

By power required & energy, are you essentially asking about fuel efficiency (over distance / time)? So which uses less fuel, a regular jet vs a Harrier? Or a regular plane vs a helicopter?

Very few aircraft have thrust-to-weight ratio larger than 1, i.e. have engines powerful enough to lift aircraft weight alone. For example, F-15 has thrust-to-weight "barely" 1.07 and most airliners are closer to 0.2

@el.pescado: And what is the fuel burn when your F-15 is at full throttle, vs normal cruise?  Or a Harrier doing a vertical takeoff (or hovering) vs maintaining altitude with wings AND overcoming the drag needed to cruise at several hundred knots?

Some intuition: Flying machines that are entirely wingless exist. These are called rockets. They typically have a lot of problems with endurance and terrible loiter time. Also, on the other extreme, you have things like engineless gliders which can fly despite having no engine at all - so nothing to point down.

I asked a similar question on Physics.SE a while back: https://physics.stackexchange.com/questions/122102/what-does-a-wing-do-that-an-engine-cant . The tl;dr: energy goes up as v^2, while momentum goes up a v. For the same momentum, you can either move a little air quickly, or a lot of air slowly; the first of those requires more energy, because of that ^2. Engines do the first option, and wings translate that energy it into the latter (and thus get more momentum for the energy).

The very simple answer is to observe that commercial airliners have wings, which weigh and cost.  Commercial airliners are designed to minimize the cost, so if they could reasonably get rid of the wings they would.

Interesting question. Purely empirically, it is the lift-to-drag ratio you are looking for. If you take this value as given for any particular aircraft, you have a direct answer for how much more effective wings are. It is the ratio of the lift to the total drag. The engine only needs to overcome the drag.
With L/D equal to unity you would need the same thrust as for the vertical take off. But even quite "bad" fixed wing aircraft would have L/D about 5. Gliders or similar aircraft built with a strong focus on aerodynamics can have an L/D of 50 or more (at least in some narrow range of airspeeds).
So yes, wings are more efficient. About one order of magnitude as a rule of thumb for common aircraft and optimal airspeed.
Why your reasoning with air pushed down is incorrect is more tricky to explain. I'll start with the assumption that, as air passes an airfoil, its speed relative to the airfoil is unaltered, and only the direction changes. (I know air slows down at least because of friction etc., but these are, at least theoretically, avoidable things not related directly to creating the lift. If there is something intrinsically related to lift which makes airflow change not only direction but speed too, then someone will hopefully correct me here.)

See the image. Airmass moving initially towards the airfoil with velocity $\vec{v_0}$ is deflected down by angle $\alpha$. Therefore the change in the velocity is $\vec{\Delta v}$. This change can be divided into horizontal and vertical components. To hold the aircraft in the air, the vertical component has to be equivalent to the aircraft's weight divided by the mass flow rate over the wing. The vertical component is related to the horizontal by
$$
\Delta v_{\rm horiz}=\Delta v_{\rm vert} \cdot \tan{\alpha\over 2}.
$$
So, from this simplistic view, the drag would be $\tan\alpha/2$ times the lift. Higher mass flow rate over the wing (longer wings, higher airspeed) makes it possible to keep the same lift with lower deflection ($\alpha$), thus less drag due to the generated lift.

Additional comment: how is it related to power and energy
The answer above is focused on how wings decrease necessary engine thrust, but original question could be interpreted in terms of energy efficiency too. I'll try to add some comments on this part.

simple example – rocket engine: not too typical for aircraft, but simple one. Rocket consumes same amount of fuel per second in order to generate a unit of thurst regardless of its size and regardless if it is pointing upward (and is static with respect to air) or forward (and moving through the air). You need to burn proportionally more fuel per second to generate higher thrust. So, for the rocket propulsion, you will save fuel in the same ratio as the necessary thrust decreases.
Propeller or jet engines are more complicate as their thrust and fuel consumption depends on engine movement through the air too. As David K pointed out in his answer, we can use momentum and kinetic energy of accelerated air to get power needed for unit of thrust.
With some simplifications, thrust is mass flow rate through the engine/prop multiplied by change of flow speed it causes. $T = \dot m \cdot (v_{\rm out} - v_{\rm in}) = \dot m \Delta v$. Power needed for this is $P = \dot m \cdot {1\over2}(v_{\rm out}^2 - v_{\rm in}^2)=\dot m\Delta v\cdot(v_{\rm in} + {\Delta v\over 2})$. Thus
$$
{P\over T}=v_{\rm in} + {\Delta v\over 2}\,.
$$
For stationary engine holding against gravity, higher thrust compared to flying fixed wing aircraft is needed as shown above. If we do not "cheat" by increasing mass flow rate through engine (like making it helicopter rotor or using multiple engines), $\Delta v$ has to be increased in  order to achieve the necessary thrust. So you not only need more power because of increased thrust but also more power because of increased Watts per unit of thrust. Note that even "helicopter cheat" does not work too well. To match power consumption of engine generating less thrust thanks to the wing's L/D you need to improve P/T too – by decreasing $\Delta v$, thus increasing mass flow (rotor/propeller radius even more than proportionally to increased thrust).
What about decrease of P/T due to the movement through air? Well, it depends on particular engine and its $\Delta v$. It will be typically in similar order of magnitude as airspeed (or even less), so we can not neglect $v_{\rm in}$ in watts-per-thrust equation above. There is a efficiency penalty when engine works on moving aircraft. But it should be still worth of it as the gain provided by lift is greater.
A simplified example: We have an engine capable of producing enough thrust to lift aircraft vertically. It can be throttled by changing $\Delta v$ without any practical problems or change in its internal efficiency. And let's assume that mass flow rate through it is fixed area $S$ multiplied by air density and multiplied by arithmetical average of speeds of entering and exiting air. For hovering aircraft and stationary engine producing thrust equal to aircraft's weight, $w$ it is
$$w=\dot m \Delta v_{\rm hover} = \rho S \Delta v_{\rm hover}^2 / 2\,;\quad \Delta v_{\rm hover}=\sqrt{2 w \over \rho S}$$
and thus
$$P_{\rm hover}=w\cdot \Delta v_{\rm hover}/2=\sqrt{w^3\over 2\rho S}\,.$$
The same aircraft flying on its wings does only need $w\over L/D$ of thrust. Airspeed is $v_{\rm air}$. Equation for the thrust: ${w\over L/D} = \dot m \Delta v_{\rm flight} = \rho S \cdot (v_{\rm air}+{\Delta v_{\rm flight} \over 2}) \cdot \Delta v_{\rm flight}$. Thus
$$\Delta v_{\rm flight}=\sqrt{{2w\over(L/D)\rho S}+v_{\rm air}^2}-v_{\rm air}$$
and
$$
P_{\rm flight}={w\over L/D}\cdot(\sqrt{{w\over 2(L/D)\rho S}+{v_{\rm air}^2\over 4}}+{v_{\rm air}\over 2})\,.
$$
Unfortunately, I do not see any way how to simplify and compare $P_{\rm hover}$ and $P_{\rm flight}$ so some concrete numbers:

Light aircraft, 1 ton, 100 knots, $S = 5\,\rm m^2$, $L/D = 15$: $P_{\rm hover} = 290\,\rm kW$, $P_{\rm flight} = 35\,\rm kW$.
Heavy aircraft, 100 ton, 200 knots, $S = 50\,\rm m^2$, $L/D = 15$: $P_{\rm hover} = 90\,\rm MW$, $P_{\rm flight} = 7\,\rm MW$.

So, based on these simplifications flying with wings with similar kind of engine should be significantly more efficient in energy terms too. And, additionally, you are already moving forward using power $P_{\rm flight}$. For vertical engine extra power would be necessary for overcoming air drag due to the movement.

As interesting as this is, It does not quite provide a complete answer for me. Even with a lower deflection the increased mass being deflected by a longer wing or higher airspeed would cause the drag to increase significantly. As this is intrinsically related to lift I feel it should be taken into account here, however it should be noted that I have no idea what I'm talking about.

Well, the relation between vertical and horizontal speed component is the same as relation between airflow-bending-induced forces in respective direction, regardless of magnitude of v_0. Vertical force is fixed (weight of aircraft), so horizontal force decreases with decreasing alpha. I can add formula how alpha changes with v0 into my answer if you are missing just that. Other drag forces not related to pushing air downwards are important in praxis too, of course, especially if you want to make fast aircraft with high L/D, but I am afraid, there is no simple theory or equation for these.

You are absolutely right, I had something twisted around in my head, re-reading your answer (several times) sorted it out. Thanks!

I don't totally agree with using the L/D ratio to say that wings are more efficient. A large L/D ratio means that most of the engine's thrust ends up being used to generate lift and sustain the cruising altitude, whereas only a small fraction of the thrust is actually used to sustain the cruising speed. But the wing itself adds drag to the airplane. So my reasoning is that if you remove the wing and re-orient the thrust of your engine downwards, you would probably reduce D by a lot but while still generating the same amount of lift. This would require (slightly) less power.

@user9037 I do not quite agree with this interpretation of L/D. L/D says that aircraft producing x units of lift (typically weight of aircraft) flying through air does produce x/(L/D) units of drag in the same moment. No engine involved so far. Lift holds you up (better: holds const. vertical speed, be it zero or whichever other number) because it cancels with weight. To cancel the drag you need an engine thrust. And only for this. So to hold 1 ton heavy aircraft with L/D=10 flying, you need 1 kN of thurst. To lift it vertically 10 kN is needed. Wing drag etc. is already included in total L/D.

OTOH my answer is more about the thrust necessary than the power, which are bit different things. Maybe David K's answer is better focused on what are you asking.

Little nit-pick -- lift does require viscosity, it is absolutely essential. The "theoretical" argument that it doesn't isn't based on theory so much as mathematical approximations and careful selection of domains. Doesn't change your answer any, but it is a subtle point that is very often missed. I've written about it over on Physics.SE.

@Martin: Thanks for your clarifications. It starts making more sense to me. I still need to wrap my head around the energy aspect though.

@Martin: Your extra comment makes it clear to me how the wing ends up being more energy-efficient. Many other answers also touched upon important aspects of this question, but I think yours now has a more complete mathematical explanation so I'll mark it as the selected answer. Thanks to everyone for their contributions!

In a traditional aircraft the majority of the power from the engine is used to keep the aircraft moving forward at a certain speed. Very little of that power is actually needed to create lift.
Consider a simple paper airplane. It flies for a long time with no engine at all, until the drag on it causes it to slow down and if loses lift and descends to the floor.

In the hands of a skilled pilot, gliders can stay aloft for hours with no engine at all.
I'm not going to get into the argument about whether wings work by directing air downwards or not because it is simply irrelevant. The basic truth is, when a wing is oriented in a way that it provides lift when moving forward, all you need the engine to do is drive that wing, and the rest of the aircraft, forward at that speed.
The wing and the body of the aircraft create an effective drag as they are pulled, or pushed, forward and the engine needs only to create that amount of force so as not to slow down. That force is A LOT less than you need to lift it directly.
Most aircraft engines simply do no have the thrust force to lift the aircraft on their own. During early aircraft development many attempts were made to do so and failed because engines of sufficient strength simply were not available.

Wings had been around a long time before the Wright brothers came along, but flight was unpredictable and uncontrolled. The first true aircraft was invented because the brothers discovered and invented a mechanism to allow them to control the wing(s).

In short, it is far easier to provide lift with wings than by using thrust vectoring.
HOWEVER: At this point you are probably still scratching your head wondering how you can lift an aircraft without actually getting that amount of power from the engine.... So let me try to explain.
Let's say you have a car, and I tell you to lift it 6 feet... Well, unless you are this guy it's just not going to happen...

But what about if you do the following?

Well, you might complain and be out of breath, but you can see how, if the ramp were a long enough slope, you could use our muscles to get the car up to that height.
Because we are slow-moving creatures we think of air as nothing at all. However, air becomes a different thing when you try to move it out of the way very quickly. It becomes significantly "hard".
An aircraft in flight can therefore be thought of climbing an air ramp as shown below.

The aircraft and wings cut through the air reasonably easily, but the air under the wings, and body, acts like a ramp. The bigger the wings, the harder and more solid the ramp. This provides the lift.. keeping the aircraft up.
Of course the ramp is not solid, and effectively drops as we push the aircraft forward. In other words the aircraft is dropping and climbing at the same time. When in level flight the ramp is dropping at the same rate as the aircraft is climbing it.
What that means is the wings give you the mechanical advantage of using a ramp to reduce the force needed to perform work. Ignoring drag, the work required is the same as if you lifted it vertically, but since you spread the work over a long forward distance the effort required from the engine is divided significantly.
Efficiency:
Now is that more efficient? Well, traditionally ramps and other mechanical advantage devices are less efficient than a straight lift because there are losses involved in extra friction in the apparatus.
However, vertical propulsion based lift systems themselves are horrendously inefficient.
As we discussed about, air is harder to move the faster you try to move it. This means doubling the power of the engine does NOT translate into doubling the thrust, it is more of a exponential function. That is, you need to burn more than twice as much gas to get double the thrust.

Worse, for any given engine, there is a limit to how much thrust it can produce. Eventually the air cavitates in front of it. It will, if it can turn fast enough, suck ALL of the air from the intake so fast that a vacuum forms. At that point the engine is starved of air and can go no faster no matter HOW MUCH fuel you pump in. That means, in order to get more thrust, you need a bigger engine, which means more weight, which means you need more thrust.... Do you see where I am going with this?
And remember, that is just to keep you up, you still have to use more power to go from point A to point B.
As such, even with the drag losses, winged flight still uses a lot less gas for any given journey distance.

Very nice first answer! Welcome to [aviation.se].

@FreeMan thanks :)

This is a brilliant answer - thanks! A lot of the other answers assume that the reader already understands the answer to the OP's question :-)

Nice analogy with elevating a car.

With regard to energy expenditure and power,
for a given amount of force that is to be produced by accelerating
an air mass,
more power is required when you accelerate a small air mass
in each period of time than when
you accelerate a large air mass.
This is because the force is proportional to the change in momentum
of the air mass, whereas the power is proportional
to the change in kinetic energy;
and while momentum is $mv,$ kinetic energy is $\frac12 mv^2.$
The typical airplane engine grabs relatively small parcels of air
and propels them backward at high velocity. A large propeller,
or a high-bypass turbofan with a large intake, will do better than
a small propeller or a turbojet with a small intake.
But the wing of a typical conventional aircraft "grabs" a much larger parcel of air during any unit of time than its engines do.
By pushing the wing forward through the air, the aircraft converts the relatively inefficient production of force by its engines (taking small parcels of air from in front of the aircraft and accelerating them
rapidly backward)
into the much more efficient production of force by its wings
(taking large parcels of air above the aircraft and accelerating them
relatively slowly downward).
Simply turning the typical engine (jet or propeller) of a conventional aircraft downward
does not allow the aircraft to accelerate nearly as much air downward
as the wing can when the aircraft is in normal flight.
In a helicopter (also known as a "rotary wing aircraft")
the engine turns the wing
(aka rotor), thereby pushing it through the air and accelerating air
above the aircraft downward
whether the fuselage is moving forward through the air mass or not.
A helicopter can therefore take off vertically with a relatively
small powerplant compared to what you would need to take off vertically
with anything like a conventional fixed-wing aircraft engine.
If you think of a helicopter rotor as a "downward-pointing fan,"
then it actually works reasonably well.

Thanks for pointing out the difference between kinetic energy and momentum, and why a large wing is advantageous. There is still something that bothers me though: the wing does not produce force, it converts force. Specifically, it converts the horizontal thrust of the engine to vertical lift. It does so efficiently, but not with more than 100% efficiency. It cannot compensate for the engine being an inefficient high-speed accelerator of small air masses (right?). So why would it be less efficient to point the thrust downwards directly (like in a helicopter)?

It is quite crucial to differentiate between force (thrust) and power (energy) here. Wings "convert" force with much higher "efficiency" than 100 %, if I would use such wording. In the same way as simple pulley system can multiply force ("convert" it with more than 100% "efficiency").

Actually, pulley system is probably nice analogy here. Your body is (biologically) inefficient producing extremely large forces. But you can can use pulley system to exchange force for greater length of rope ... which can turn out to be more efficient even if you lose some energy in the pulley system. In the same way wings allow engine to operate on lower thrust. Saving energy indirectly because it is better for a "normal" engine. Of course, as David pointed out, you could scale engine to provide higher thrust efficiently, which will work, but ends up with something like helicopter.

@user9037 I am not sure what you mean by saying a wing 'converts' force, but you appear to be using it synonymously with 'redirects', which is not so. A moving wing experiences a force, from its interaction with the airflow, which can be resolved into lift and drag. In order to keep moving at a constant speed, the propeller only has to generate enough thrust to overcome the drag, which, for any half-decent wing, is much less than the lift component.

But helicoptors can't take off vertically if they are loaded near their maximums. They need to be moving forward, which enables them to produce more lift. How's that work?

@PhilFrost In this answer I was trying to get at the first-order effects. Hovering entails some additional complications that I think are best answered in another question--and in fact I think they are answered in response to https://aviation.stackexchange.com/questions/22427/why-dont-helicopters-always-takeoff-from-a-hover-out-of-ground-effect

Sure, but I meant it as a rhetorical question. Given those complications in hovering, can we really think of a helicopter as a "downward-pointing fan"? Perhaps not, at least in the way the OP would think of it. There still must be some explanation for how moving forward generates lift more efficiently besides the mass of air being redirected.

@PhilFrost I was under the impression that a helicopter can still lift its own weight and some load (though not its maximum) vertically. Is that a misapprehension? Whereas if we just take the propeller(s) of an ordinary fixed-wing aircraft and point them downward, at full power you'll barely be relieving some weight from the wheels. The comparison of a hovering rotor with a wing moving forward through undisturbed air isn't perfect, but the point really was that if you point a fan down, it needs to be a really big fan moving relatively slowly.

I think your impression is accurate. I was able to find the name of this phenomenon: translational lift.

@Martin OK that makes a lot of sense. So the engines provide some kind of lever effect that allows the airplane to fly with much less thrust than its weight would require.

@user9037 Wings work as a lever, kind of, allowing flight with lower thrust, yes.  Second part to this is that aircraft's (reaction) engines do consume power and fuel even when generating thrust while stationary (in contrast to, for example, car on a road, which does not need, at least theoretically, any power until it starts to move), so decreasing thrust needed pays off here.

If we ignore the losses, maintaining the aircraft at a given altitude requires no power, since no work is done on it. It does, however, require a force, and you seem to confuse force and power. The term efficiency has no meaning (at least no well-defined meaning) when talking about forces.
For example, I can hold a weight of 20 kg in my hand, and I could hold 200 kg using a 1:10 lever. Sure, you can say that the lever is 10 times more efficient, and in that sense wings are more efficient than vertical takeoff engines: you can take off with an engine which has 10 times less thrust. The corollary is that you will need 10x more time to reach a given altitude, just like I would lift a 200 kg weight 10 times slower using a lever than I would lift a 20 kg weight with my hands.

Maintaining the altitude requires no work, however generating the lift force should require energy since air needs to be displaced. It is not a static equilibrium. So, since energy is consumed, the question is whether the wing is somehow more efficient than a downwards-pointing engine. Am I missing something?

Uhm, yes, the wing is more efficient because it generates the same amount of lift with less engine thrust, and as you said yourself generating thrust consumes energy. I'm simply pointing out that the confusion comes from a sentence the power required to keep an airplane at a constant altitude [is the same for the wing and the engine]. The power doesn't have to be the same, since no real work is done.

Okay, I understand your point now (the power doesn't have to be the same, since no real work is done). Thanks for clarifying this!

A force equal to the force applied by gravity must be generated somehow to keep the plane in the air. But there's no rule that requires some particular amount of energy to be expended to apply that force. In theory, the force could be applied with no energy expended at all. For example, a permanent magnet pushing against the Earth's magnetic field could (in theory, anyway) do it.

@DavidSchwartz In theory, simply going fast enough (to develop enough centrifugal force) would be sufficient, but yeah, that's my point.

@DavidSchwartz Even more simply, if you were to just lift an object and put it onto a table it happily remains at the height of the table without any further input of work.  I think this answer is great - it hits directly at the misconception that OP has (and that the other answers have missed).  For simply the cost of drag a wing produces lift that, in some limit, makes the aircraft/air system approach that of the object on the table.

Skimming through the answers, I'm missing a very simple approach to explain the difference:
Listing the inefficiencies for both design solutions
Fixed wing

non-lift-producing drag - Some of the drag is not related to producing lift, e.g. friction of air on wing surface.
wing tip vortices - The pressure difference between above and below the wing seeks to equalize by a flow of air from below to above. This can be mitigated by winglets and/or high aspect ratio of the wing.
anything - This is a placeholder for anything, I might've missed. See below for its counterpart.

Rotary wing

non-lift-producing drag - Same as the equivalent point above, except for different wing profile, airfoil and varying airspeeds. See below for more on varying airspeeds.
wing tip vortices - Same as the equivalent point above, except wing length and thus aspect ratio is more limited in design choice. Winglets would cause lots of structural problems and increase drag over-proportionally, because they are by definition on the (fast moving) tip.
anything - Basically anything that applies to fixed wings does apply for rotary wings, too. Plus, often times the non-uniform airflow (see below) does not allow for optimizations in wing profile, airfoil, etc.
non-uniform distribution of air speeds over the blade - The tips of a rotor are moving faster through air than its base. Thus optimal airspeed is hard to obtain everywhere on the rotor blade at once.
different airspeeds for preceding vs. receding blade - Forward speed of the aircraft is added to the air speed over the preceding blade but subtracted from the receding. This difference adds to the problem of achieving optimal airflow.
need for counter torque - The tail rotor in the classic helicopter design requires power from the main engine without adding to lift or forward thrust. It's basically a "useless necessity". Twin rotor designs may suffer from increasingly "disturbed airflow" (see below).
circular motion - Circular motion is basically accelerating towards the center. "Flying straight" would be more efficient, e.g. there are bearings on the rotor that leak the angular momentum. In comparison a fixed wing doesn't leak its momentum besides the other inefficiencies. This also poses structural requirements on the rotor blades that may limit other design optimizations.
disturbed airflow - The previously receding blade moves through wake of the previously preceding in the next revolution. The disturbed air doesn't create as clean an airflow as it would undisturbed. This reduces lift to drag ratio.
unoptimized airfoil - Multiple points above already mention this, either by making it necessary (structural requirements in circular motion) or by preventing optimization (non-uniform distribution of air speeds over the blade).

To come back to the original question:
As a rule of thumb, we can assume that the longer the list of inefficiencies the less efficient the design. Especially when everything (and anything) on the one list appears on the other, too. There would need to be big qualitative differences in each point for the rule of thumb to be violated.

The way "basic" flight was always explained to me is that because of the shape of the wing, air over the top has further to go, thus is "stretched" the air under the wing has less distance to travel. So the air under the wing has more "pressure" then over the wing. At no time are you actually "pushing air down". At least not exactly. The weight of the craft does cause the air under the wing to "smoosh" (or displace) the   same way a boat does, while there is no force (aside from the lift) pushing "up" on the wing.
This is all a very simple explanation. But the core part, the very very important part is that in no aircraft, fixed wing (plane) or rotary (helicopter) is ANY lift generated by pushing air down. Lift is generated by there being less air pressure "on top" of the wing then there is "under" the wing when combined with the downward force of gravity. It's the pull downward that actually makes planes go up, as odd as that sounds.
Now in your question you want to know why it takes less energy to fly "plane like" then "helicopter" like. Again, remember that pushing air down doesn't do squat until you get into rocket engines.
To answer that we take a look at what each engine is trying to move. In a small plane the engine has to move a propeller. Lets say about 70 lbs. With that engine turning 70lbs of weight it can "pull" (in much the same way as the wing) a small plane about 140 knots. That is more then enough "speed" to make the wingy bits of the plane generate lift. Keep in mind "lift" doesn't have to be this huge large force, it just has to be a tiny bit stronger then gravity.
In contrast, a helicopter's "blades" (there just wings that spin around) weight around 250 lbs.  It's hard to convert the rotational speed to knots but at 650 feet/s that's roughly 385 knots (the math on that is very rough)
So, it takes far less energy to pull an aircraft forward at 140 knots. Then it does to spin a set of wings at 384 knots.
Keep in wind that the wings on a plane can be MUCH larger then the wings on a helicopter. That extra surface area creates more lift at slower speeds.
To make it more complicated, all of the "planes" energy is used to move the craft forward. That's it. A plane only goes one direction. They don't really turn as much as they "fall" in a prescribed direction (by generating less lift on one side or the other along 3 axis). The helicopter on the other hand has to spend some of it's energy to move "forward". It's "forward" motion is basically prescribed falling just like the plane, but then has to have energy spent on generating more lift, where as the plane just moves forward.
TL;DR I'ts not really apples to apples, but it takes less energy to generate the same lift by moving forward then it does to spin wings in a circle and generate lift that way.
HUGE NOTICE I used the speeds, and flight profiles from many aircraft. The plane I used was "a Cessna"  but I took numbers where I could find them so some are the beloved 172 other are the other variants.  The helicopter numbers are even more varied. I tried to keep it to lighter weight helicopters but I may not have succeed. The important part is that the theory is right, but don't try to count on the math in any real way.
Another note some VTOL craft do actually push air down, but this is even less efficient then spinning tiny wings. In short pushing down to go  up is like a rocket, creating less pressure on top and "floating" up is a plane.

The pressure explanation and the "pushing air down" explanation are basically just two ways of describing the same thing. We all agree that the wing acting on the air accelerates the air downward. In order for that to happen, a force must be exerted on the air. And, if that happens, then an equal magnitude, opposite direction force acts on the wing. This force is the lift. The "longer distance to travel" part, however, is not really correct. Indeed, there are symmetric wings where the transit distance is equal which still fly.

I just want to add something that I feel has generally been overlooked here.  As the specific airspeed over the wing/propeller increases, the drag increases more than just linearly, its exponential. In other words, since the airflow(in mass) over a wing is a lot higher, it can produce x amount of lift at a low airspeed, while with an engine, since it has a lower airflow(again in mass) needs to move the air faster over the propellers in the engine to generate the same lift.  As the drag is not linear, it needs considerably more power to overcome the drag from the engine, which is what causes the inefficiencies.

A major advantage a wing has versus an engine is that in typical use it will constantly get to encounter relatively undisturbed air.  A downward-pointed engine would create an area of low pressure above it, and air flowing into it will be moving downward even before the plane gets to do anything with it.  The only way the plane can generate thrust will be to accelerate the already-moving air to an even higher speed.  The amount of energy required to accelerate a cubic meter of air from 9m/s to 10m/s is almost twice as great as the amount required to accelerate 10 cupic meters of air from 0m/s to 1m/s, but the amount of lift generated by the latter will be ten times as great.

Thanks to your explanation. It's more condensed than Martin's answer, but I marked his answer as valid because it has the relevant formulas.

@user9037: I judged that Martin's was a high-quality answer, but also thought it would be helpful to have a simple intuitively-clear answer as to why flying is more efficient than hovering.  It would seem conceptually that a hovering vehicle should be simpler than a flying one (and indeed the whirlygig toys that predated the airplane operate by simply directing thrust downward) but I think the need for still air is something that may not be obvious until one considers it, but once considered will immediately make many things obvious.

Wouldn't the suction which creates said low pressure above the engine also be subject to Newton's Third Law, though? If the engine is causing  a given parcel of air to get accelerated downward - whether before or after it enters the fan - is not an equal, but opposite force applied to the engine?

@reirab: Everything is subject to Newton's third law, but the plane isn't the only thing acting upon the air.  Air is supported by the planet, and if a plane were flying in large circles around a planet, most of the air it disturbs would have a chance to convey the force of the airplane to the planet before the next time the plane encounters it.

@reirab: The total amount of force conveyed to the planet through the air is going to have to equal the weight of the plane no matter how it is conveyed, but the amount of work required to impart acceleration to the air will vary depending upon how that air is already moving.

@supercat Yes, I'm not disputing those things. I was referring to your statement about the engine creating an area of low pressure above it, but seeming to suggest the downward acceleration of that air before entering the intake does not cause an upward force to be imparted on the engine/aircraft. If work is being done on the air, surely an opposite force is being imparted on the aircraft, no?

Unless, of course, you're referring to an airplane flying straight up, in which case the air would already be moving downward relative to the engine without any acceleration required and I misunderstood your meaning. And, of course, that's exactly what happens in the case of a horizontally-oriented engine flying forward through the air.

@reirab: I guess I drifted a bit from my main point into issues related to hovering which aren't necessarily applicable when climbing straight vertically fast enough to keep drawing "fresh" air.  The aircraft does effectively gain "momentum" from air it causes to be moves through the engine, but it takes more kinetic energy to transfer a certain mass of air through the engine at a high speed than to achieve the same transfer of momentum by accelerating a larger quantity of air to a slower speed.

@supercat Yeah, that part is definitely correct.

Addng to @Dmitry Gregoriev excellent answer: it could be that your question boils down to: why is a fixed wing more efficient than a disk shaped wing.
Because of the lifting line theory. Creating a given amount of lift over a finite span is the more efficient the larger the span.

Let's go back in time to ask the pioneers attempting human powered flight. The wing design offered efficiency over designs that relied on vertical thrust. Such efficiency encouraged further research.
The lift force developed by a wing at proper airspeed relies on pressure. Pressure is also responsible for the lift (buoyancy) force in boats (water pressure instead of air). You can ask a submarine designer about abandoning ballasts and adding downward propellers to maintain depth.
Although a different mechanism, a hot air balloon can demonstrate that different amounts of energy is needed to produce the same lift using a different engineering principle. A small jet engine can actually be used to generate enough hot air to lift the balloon with carriage. However, pointing that small engine downward will not provide enough thrust for equivalent lift.

Regarding the article:
You say that your understanding is that the article says that wings on an airplane basically create lift by "pushing" air down.
This statement is incorrect to the extent that this implies that lift is created only by the bottom of the wing.  (I am not 100% sure that is what the article is saying since it merely states that "Planes fly by throwing air down.")
The theory that lift is generated only by the bottom of the wing is referred to in the NASA webpage described in earlier comments as the "Skipping Stone" theory. The webpage states that this theory is incorrect because it does not recognize that lift is also created by the top of the wing.
Another NASA webpage explains that "Lift is a force generated by turning a moving fluid" and that "Both windward [bottom] and leeward [top] parts deflect a flow." (words in square brackets added)
Let me add some other evidence and observations:
The book "Stick and Rudder" written by Wolfgang Langewiesche in 1944 has sometimes been cited in support of the "Skipping Stone" theory because the book talks about "slamming" the air downward. However, in his explanation, the author states that:
"The main fact of all heavier-than-air flight is this: the wing keeps the airplane up by pushing the air down. .... It shoves the air down with its bottom surface, and it pulls the air down with its top surface; the latter action is the more important." [p. 9]
Another observation that highlights the importance of lift created by the top surface is the fact that a stall involves the loss of lift produced only by the top surface. The bottom of the wing continues to produce lift, as before. But it does not provide enough lift alone to keep the airplane aloft.
The importance of lift generated by the upper surface is also familiar to any pilot who has flown into icing conditions. As noted in "Aircraft Icing" AOPA Safety Advisor, Weather No. 1, p. 2:
"Wind tunnel and flight tests have shown that frost, snow, and ice accumulations (on the leading edge or upper surface of the wing) no thicker or rougher than a piece of coarse sandpaper can reduce lift by 30 percent and increase drag up to 40 percent. Larger accretions can reduce lift even more and can increase drag by 80 percent or more."
I also recall having read a couple of anecdotal accounts of what happens when there is interference to the air flowing across the upper wing, but can't provide a cite. One involved Martin Caiden who set a record for number of skydivers jumping at one time when 19 skydivers were hanging along the top of the wing of his Ju 52 "Iron Annie".
I do know that business jet manufacturers used to have to add strakes on the top of their airplane wings to insure that the air along the top surface flowed as originally intended. The engineers at Cessna were quite proud when they were able to design a wing (the Citation X) which did not require any strakes.

Isn't this a bit like comparing apples and oranges? Given no outside forces, wings without an engine to drive them do very little to get you into the air.
An engine can create 'lift' by pointing it downwards. To get off the ground the engine would have to generate thrust to counteract its weight. If you add wings, you can get into the air with far less thrust. So wings INCREASE the efficiency of an engine when it comes to how much thrust is required to become airborne.

If you add wings, you can get into the air with far less thrust. Yes, but why?

The article makes a misleading generalization, as it was one of the first things I learned during flight training. Airplanes don't fly by "throwing air down", they work by generating reduced air pressure that lifts the airplane up (hence, "lift") and forward (thrust). Both wings and propellers (and turbines) are airfoils whose upper curved surface accelerates air as the airfoil moves through it, thereby reducing air pressure. The relatively decreased air pressure above the wings and relatively increased air pressure below the wings and behind the propeller floats the airplane up and forward.
A modicum of truth is that the air flowing over the wing is deflected downward, and some air is compressed beneath the wing as the airplane moves, but is a much smaller component of what keeps the airplane flying.
Lower air pressure behind the wing produces drag and is a byproduct of lift. Surface friction of the lifting surfaces and body of the airplane are also components of drag. A vortex generated at the wingtip as the higher- and lower-pressure airflows converge and spiral around each other can also be a strong component of drag, as well as causing turbulence that can affect other aircraft.
To address another part of your question, producing lift and thrust follow the same principles using conventional piston and jet engines (rocket engines produce thrust by expanding gases). Perhaps one of the best visible examples is the Osprey tilt-rotor aircraft with large propellers that can produce lift, thrust, and any combination in between, depending on the angle of the engines.
@ymb1's analogy about pushing a box was an excellent choice. Moving perpendicular to gravity (i.e. moving wing forward) takes less force than simply opposing it (i.e. thrusting downward). So, wings are a more efficient choice, from both a structural and complexity perspective.

So what happens to the air flowing over the wing? The suction on the upper surface pulls the air above the wing down and the pressure on the lower side pushes more air down. As a result, the air is accelerated downwards when it flows off the wing. Yes, airplanes fly by pushing air down; this is not a simplification but a consequence of the pressure field around a wing or helicopter rotor.

You might want to read up on the theory of lift.  The high pressure below, low pressure above theory has been well and truly laid to rest.  Lift is the opposite and equal reaction to the air being accelerated downwards.

@Simon: Thick airfoils have suction on both sides, but there is always a pressure difference when there is lift.

@PeterKämpf Yes,  I was pointing out that the high pressure "lifts the wing up" myth has been abandoned.

@Simon … yet at supersonic speeds it is partially true. And in ground effect, too.

Doesn't matter how you analyze it, some basic facts don't/can't change. Air is fluid; you push on it, it moves. Air has mass, so if you push on it and it moves, work is done. Wings never "pull" on air; if, as you say, the wing reduces air pressure on the upper surface, ambient air pressure pushes air down in response to the unbalanced forces. So, as a wing is generating lift, air is being displaced downward, which requires an input of work. You can still consider the wing as generating vertical thrust, via large masses of air accelerated to very low downward velocity.

Unfortunately, a lot of the material even that's taught to pilots in flight training is wrong in regards to how lift generation works. Both Bernoulli's Principle and Newton's Third Law are both true at the same time. It's not a matter of some of the lift being generated by one and some by the other. All of the lift follows Newton's Third Law. The force applied in the downward direction to the air is equal and opposite to the force applied to the upward force applied to the airplane. When the vertical speed is not changing, this force is equal in magnitude to the airplane's weight.

@PhilipRoe What would be news to most aerodynamicists? Having worked with several of them at an aerodynamics testing facility, I'm pretty sure they're generally aware of Newton's Third Law. Maybe you meant to reply to Simon instead?

@reirab: Oops, apologies. I did indeed mean to reply to Simon.

Airplanes do fly by throwing air down. A flat plate can create lift.

Wings are an economical way to accelerate a mass of air downwards. The fact that natural selection has chosen that system over other alternatives possibly means that wings are the most economical solution...

natural selection has no much bearing in airplane manufacturing. or we would have aircraft that have to flap their wings. or birds with internal combustion engines.

@Federico  It's the other way round... We have not (yet) been able to build neither viable ornithopters nor engines that convert chemical energy into mechanical work. Natural selection is always right...

natural selection is not magic, it works with what it has, that more often than not is quite sub-optimal, both in the starting point and in the final outcome. its only "goal" it to make living being X capable of surviving living being "Y". see also here: https://biology.stackexchange.com/q/21272/9247

@ Not sure it's 'sub optimal'... After all, it has been able to produce a 'Federico'. Do you know of any alternative, better way to build one...?

plenty. I have plenty of physical defects, as we all do. starting from this little pesky proclivity to get ill.

Natural selection arguments aside, "it is X because X is the best" is not a particularly good answer for any question on an SE site. We're really looking for an answer that describes why X is the best.

@Federico  Well, I feel really sorry for your shortcomings, but that's the problem with reality. Fantasy is another thing... Sure you are able to manufacture an 'improved Federico' in your imagination. But making a real one you can't... You don't know of any alternative way to produce a run-of-the-mill 'Federico', much less an 'improved Federico'. Natural selection has, however, made one that works... And that's a fact, not fantasy...

@FreeMan Natural selection does always win. Is an adaptation the best? Well, were it not the best, there would be another, better adaptation, and that adaptation would be necessarily the best one...

Sadly, "natural selection" (a biological process, not a mechanical engineering one), doesn't always win. There are a lot of things that happen in my workplace that are in no way the best way of doing things, but "that's the way we've always done it" is the reason for not changing. Therefore, your argument of "Wings are best because they are" would be an accepted, but very invalid answer in my workplace. I will restate that your "X is best because that's the way it's done" argument is invalid without supporting evidence. However, I see your mind is made up, so I'll quit now.

@FreeMan: Give the newbies some slack here, please. If this would be rephrased as "Occam's razor", the community would probably judge less harshly. xxavier has a point, even though this is no absolute proof. I have enough experience with genetic algorithms to have a healthy respect for natural selection. I hope you can be convinced that such harsh downvoting will drive new contributors away. We should not become like Wikipedia!

Natural selection has created wings which are substantially different from aircraft wings. Airplanes don't have to hide from predators, don't build nests and don't use their wings to perform mating dances.

@Dmitry Grigoryev  That's true, but in spite of some differences, they are wings after all... All flying animals use wings, and not –for instance–
downward-directed blowers, and not one has opted to be a flying aerostat, a lighter-than-air system...

@xxavier Yet aerostats have infinite efficiency at maintaining altitude: they require no thrust at all to float. You seem to defeat your own argument here.

@Dmitry Grigoryev Just see how many people are flying in aerostats and how many are flying in winged, heavier-than-air machines. That's the 'ratio of success'... Fantasy is one thing, and reality is another...

@xxavier The question is about efficiency at maintaining altitude, not 'index of success'. Not many people fly gliders either.

@Dmitry Grigoryev. It's you who's saying that the question is about 'efficiency in maintaining altitude'. That's only a part of the problem of flying, and not the most important one, as the 'success' of airships may attest...

@PeterKämpf let's not mix algorithms, designed to give a certain optimal output, with a random process that is nether designed neither tailored to optimize, but only to make the "host" survive long enough. The "cost function" is completely different.

Interesting "out of the box" answer

@Federico: Genetic algorithms use randomized variations, so they work just like evolution. But maybe you are a creationist after all, so I see no point in entering a dispute. Belief and science don't match.

@PeterKämpf no, I am a punctuactionist (Gauld), and you are a gradualist (Dawkins).

@Federico: Nonsense - change depends on environmental pressure, not on some worldview which is described by a complicated word. Change happens when it is needed.

@PeterKämpf then you are a punctuationist, but then I don't understand your view of evolution, since algorithms are always under "environmental pressure", nature less so (as you just said) and such pressure is not constant in time.

@Federico Yes, the algorithm compresses time. It does so anyway in a simulation - this is one of the points of running simulations. This does not make its optimization strategy any different from natural evolution.

@PeterKämpf "not constant in time" as in "not constant in direction nor magnitude": unless you have a cost function that changes during the simulation (but then you are not really doing engineering work) you are not simulating nature. predators change, climate changes, your prey changes. aerodynamics and thermodynamics do not change. nature "optimizes" for all of these, an engineer only the last two.


text:
Which Aircraft Utilise Fly-By-Light Controls?
Although I can find reference to the Eurocopter EC135 being a test bed for fly-by-light, I am not sure if it has entered production with the system [1]. The only confirmed aircraft I can find is the Kawasaki P-1 [2]. I am aware that the F-22, for example, has an optical fibre databus which is used to handle the large volume of data from the RADAR [3]. I am also aware of numerous test beds (the F-18 flown by NASA, the F-15 used in the FLASH program, and many others, but would welcome comments about these in case I have missed any). I am more interested in production aircraft that utilise optical fibre data links for flight control signals. For context, I was told in passing by a flight test engineer that given MIL-STD-1773 exists [4], fibre optic data links are effectively "everywhere". Based on my knowledge this is not correct, and after some research I agree with my assertion; however, maybe someone else has some additional knowledge.
Which production aircraft utilise a fly-by-light control system (other than the Kawasaki P-1)?
3 Spitzer, C. R. (2000). Digital Avionics Handbook. Taylor & Francis. Google Books
4 See for example, MIL-STD-1553 & 1773 DATA BUS PDF (66 kB)

The A-220 nee C Series is just about the most recent clean slate civil transport to be certified, and it is still FBW, although it is digital. It took a long time for digital FBW to make it from military a/c into civil transports.  I can only imagine how massive and expensive a project it would be to certify a fibre-optic data transmission system for primary flight controls on a civil transport. I would expect the same kind of delay in getting the technology from military like the Kawasaki P1 to civilian use.

Your question is already so well researched that you might have trouble finding a better answer :D

Thanks John and U. Maybe, just maybe, there are some defence maintenance people that can comment, or even someone who has worked on the EC135. But yes, my literature folder on the topic is now 203 articles on the topic (that I need to read). That sounds like a lot, but the number with the same authors providing annual updates, and to 2 or 3 conferences per year means it is still not that dense in terms of actual aircraft. Although, I still have open the scholar tab for ADOCS, "Advanced Digital Optical Control system", which used a UH-60 Black-Hawk, which I had not previously heard of.

@AerospaceDoctor, I work on the EC135 ACT/FHS Fly-by-wire/Fly-by-light helicopter as a flight control law research engineer. This system is used as the primary flight control transmission path, so it IS fully certified. But I am not sure how to answer your question: The ACT/FHS is a prototype aircraft, the "Light" part is also very much a prototype. It is  sometimes difficult to maintain. It did not go into production. I dont know why it did not go into production. But FBW is already expensive an seldomly found on helicopters, FBL even much more so.

I am just a computer programmer, with no aviation expertise. But since pretty much every optical digital system still needs wires on the ends - i.e., convert electric signals to optical and then back again at the other end, it isn't an automatic "do this" thing even if it seems like a nice idea. The three situations in general that justify fiber optics over wires are: long distance - not applicable in an airplane (even standard Ethernet can handle 328', which is longer than an A380 or a C5); security (not applicable as the systems are standalone and not part of the global wired internet);

and resistance to electrical interference - and an airplane is a relatively controlled environment so that can be handled pretty well through other means. In other words, the advantages that justify fiber optics in big buildings (or between buildings) or in power plants or other places that have lots of interference just don't apply to an airplane. And therefore why add complexity with very little benefit.

All I know is the F35 mission systems pukes spent a lot of time fussing with their fiber. Glad we didn't have to deal with that on the flight controls side.

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text:
Does lift generation create a vacuum over an airfoil?
According to Bernoulli’s principle, airflow speeds up over an airfoil which decreases the pressure, and I read somewhere that when airflow speeds up over an airfoil, this creates a vacuum. Is this statement true?
How would a vacuum above an airfoil be different from airflow separation?

See Chapter 3 here  http://www.av8n.com/how/#contents.  True, there may be some vacuum formed to suck the plane up in the air, but I think that is more than overcom by the air being pushed down by the wing, and conversely the air pushing the wing up in elevation.  And of course, the many related topics over on the right.

@CrossRoads Okay now if there is a vacuum over the wing, wouldn’t be the same as airflown separation ?

See 3.7 and Figure 3.11. The low pressure (vacuum) area can be seen to be pretty small:   The second type of stall-warning device (used on the Cessna 152, 172, and some others [my 177], not including the 182) operates on a different principle. It is sensitive to suction at the surface rather than flow along the surface. It is positioned just below the leading edge of the wing, as indicated in the right panel of figure 3.11.

At low angles of attack, the leading edge is a low-velocity, high-pressure region; at high angles of attack it becomes a high-velocity, low-pressure region. When the low-pressure region extends far enough down around the leading edge, it will suck air out of the opening. The air flows through a harmonica reed, producing an audible warning.

There are other pictures showing the air flowing up and over the wing, with separation occurring at the later part of the wing, moving forward on the wing as the angle of attach increases,  until eventually there is insufficient flow to maintain lift.

I don’t think a vacuum in its pure sense is created. A low-pressure area, yes, but no vacuum (as in: absence of air and air pressure).

Also, could I suggest rewording the title into a question, e.g. „Does lift generation create a vacuum above the airfoil?“ or similar?

@CrossRoads Thank you I will check the figures.

@Cpt Reynolds I editted the question. that is what I thought as well. even if there is a vacuum, it doesn’t contribute to generating Lift. I will check the figures suggested by CrossRoads.

@CptReynolds you have a partial vacuum, although a very weak one.  The fact that there is a partial vacuum causes a much larger package of air above the wing to be induced to move down than without.  The lift generated by a curved airfoil is so much more than a flat sheet of plywood because the airoil influences a much larger package of air as the bit of "suction" helps induce air well above the wing to move down and contribute to the action/reaction force.

@JohnK I‘m not sure I follow. A vacuum in the physical sense is the absence of matter, similar to what is found in space. I don’t think that happens on the upper surface of an airfoil. Clearly there is a low pressure area there, and of course it does massively contribute to lift. Nonetheless, there are air molecules present in that low pressure area in great numbers, so it is not a vacuum.

I'm just using "partial vacuum" as shorthand for lower pressure in one spot than in an adjacent one.  Probably not good as a scientific or engineering term.

This is not a duplicate of the question it is marked as a duplicate of.  Duplicate answers do not equal duplicate questions.

I think this is simply a terminology issue.
There is no vacuum in the physical sense on the upper surface of an airfoil. There is an area of low pressure, though, which in certain context is also called a vacuum.
A great example for this linguistic imprecision is my vacuum cleaner, inside of which there is no vacuum in its physical sense by far, but clearly it sports considerably lower pressure than the atmosphere in the rest of my living room.

I agree with you because according to "No Motion, No Lift" sub section in the following link, there has to be fluid for Lift to exist.
NASA

How much of a vacuum there is depends on flight speed. At subsonic speed the amount by which the static pressure on the top forward part of the airfoil is lowered is only a few percent of the static pressure. However, at hypersonic speed the pressure difference is substantial.
How much pressure is reduced can be seen by the lower effectiveness of conventional vertical surfaces when speed is increased. Part of that is aeroelasticity - the loads deform the structure, making the vertical less effective. But when compared with ventral fins, the pressure difference becomes obvious: The ventral fins deform as well, but their contribution to directional stability becomes dominant at higher Mach numbers because they enjoy the luxury of working in air of higher density.

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text:
Does speed or angle of attack generally have the greatest impact on total induced drag?
I know induced drag varies with speed and angle of attack in straight and level flight.  Which factor primarily determines the amount of induced drag?  The speed of the airfoil through the air, or the angle of attack of the airfoil?

The change in angle of attack --as an end result of changes in speed-- causes the greatest difference in induced drag.
Induced drag drops with increases in speed because the faster-flying wing needs to deflect air by less in order to create the same lift.
Please read this answer which tries to explain induced drag much more intuitively than what you probably have read until now. With more air flowing past the wing per unit of time at higher speed, less deflection is needed, so the backward tilt of the lift vector is smaller. Since the weight of the aircraft should not change with speed, the same aerodynamic force has a smaller backward component at higher speed or in denser air.


text:
Which country has the lowest flight time requirement for the issuance of a pilot certificate?
Which country requires the fewest number of flight hours for pilot certification?

for what kind of aircraft? typically "sport" licences or ultralight or glider licences need fewer hours than full PPLs

Some ultralights don't require licenses at all. The other issue you need to worry about is if you receive training in country X, does that license transfer to country Y without meeting country Y's minimum requirements.

If it takes you x number of hours to actually learn it, what does it matter if country A requires .8x and country B requires .75x? You will still spend x hours learning in either country.

My question was not aircraft specific so the variety of answers is appreciated. Curiosity was my motivating factor in asking the question

Here in the US you can get your sport pilot in 20 hours but that is a pretty heavily limited license, your basic VFR PPL will require at least 40 hours here in the US. You should note that this is flight time which is generally tracked on the hobbs meter (positive oil pressure or master switch on depending on configuration). There is a lot of other time in ground school, flight planing, pre-flight checks, and waiting for weather to pass that you will undoubtedly run into.
Looks like Canada requires 45 hours and thats day only. The UK also requires 45 hours.
These hours are for VFR only. An instrument rating, which is not a separate license but an additional rating on the pilot certificate, will take more hours. Some countries limit night flying privileges without an instrument rating, and if you intend on doing any serious flying its a good idea to have.
As mentioned by Ron in the comments, not all licenses are valid everywhere. Some are recognized with limitations, and some can be transferred depending on what you are trying to do. Some privileges may be recognized if you are flying to an area to visit etc. Other situations may require more training or filing different types of flight plans. Almost everyone is on ICAO format these days.

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text:
At what altitude, on land, do our ears pop?
So, it's obvious that our ears pop when we go into deep water or fly high in a plane but at what altitude on land do our ears pop? I need to know for a book.

This is pretty subjective, my ears often don't pop at all on flights. Also, when swimming, ears don't usually "pop" unless you do it purposely. It may cause the person quite a bit of pain until they figure out how to pop their ears.

My ears usually pop with the 500 ft elevation change on my drive to and from the airport from which I am based. However, they rarely pop when I am flying even with 2000+ feet of cabin pressure change.

It's more related to the rate of change of pressure, rather than the absolute pressure. If you climb an 18000ft mountain over the course of a week, your ears won't noticeably pop.

Your ears pop when your eustachian tubes equalize pressure.  This can be induced by intentional muscular action, and therefore has no set point at which they pop.  There may be some pilots who clear the pressure differential every few seconds during climbs without even thinking about it out of habit.  I can sit here in my computer chair and pop my ears  about twice per second if I try to.

Could just as equally be closed as opinion based.

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text:
Can having an incorrect hair color on my FAA medical certificate get me in trouble?
I just got a new FAA medical certificate, and I accidentally put down that my hair color is blonde when in reality it is dark brown. Also, my hair is listed as brown on my pilot certificate.
I bleach my hair in the summer, which makes it blonde.
Would such a small detail cause any sort of trouble?

This probably won't cause you any practical trouble unless you go for a new certificate or rating (the data on your medical needs to match the data on your IACRA form which needs to match your existing certificates - an examiner may refuse to conduct a test if there are discrepancies).
It's something you should correct though: It's presumed your hair color on the medical is your natural hair color, and if you get ramp checked you don't want to have to explain the discrepancy.

Counterintuitively the answer for "How do I fix this?" is buried in the IACRA FAQ:

Q:    The data on my medical certificate does not match what is in the FAA Airman Registry database, what should I do?
A:    If it is less than 60 days since the date of issue of your certificate, contact your AME (Aeromedical Examiner) for resolution of this issue. If it is 60 days or more since the date of issue of your certificate, contact the FAA Aerospace Medical Certification Division for resolution of this issue. The contact info is below.
Aerospace Medical Certification Division, AAM-300
Federal Aviation Administration Civil Aerospace Medical Institute
P.O. Box 25082 Oklahoma City, OK 73125
Tel: (405) 954-4821

Thanks.  My address is also different from my pilot cert (my pilot cert is my home address, and my medical is my school).  So I will contact my medical examiner to get both of those things changed.

got to wonder how they deal with aging pilots. Hair tends to change colour over time, adding grey and eventually going white (or disappearing completely) :)

@jwenting Well there's always hair dye and toupees :-) Seriously though: in that case your pilot certificate would become inaccurate, and there's a procedure to fix that too. (This is The Government - there's a form or procedure for EVERYTHING!)

and no doubt a fee :) Many government requirements seem designed to make you spend money on other government programs in order to meet said requirements and no other reason.

Why does the FAA even care about your hair color?


text:
How can a pilot become established on the KBIV, ILS or LOC Rwy 26 Approach?
Consider KBIV (Holland) in west Michigan:

Now consider the ILS / LOC to Rwy 26 at KBIV:

Questions:

There is a note on the IAP: "Procedure NA for arrival on PMM VOR/DME airway radials 004 CW 091" Does this mean only V55, V193, V274 or V170 or does it mean any radial between 004 to 091 off of PMM?
There is no feeder route on the plate with NoPT. Does this mean the course reversal at JAVPO is mandatory?

From experience, most of the time, it is vectors to final without a course reversal. My question is more about the charting, not the real world result, and what to do if the radar was broken that day...
How do I get to KBIV?

To answer question #1, you are not allowed to use this approach when arriving to the PMM VOR from radial degree numbers 004 ascending to 091. This means that you can not fly in to PMM from the Northish, Northeast, or East. This is probably to avoid conflict with the major congested area of Grand Rapids and its associated airport’s Class B airspace. Also, V170 is no longer an active victor airway. And, PMM R-29 is unusable for VOR only navigation. However, this does not preclude using this approach for local flights and radar vectoring where an arrival would not be necessary.
To answer question #2, the course reversal at JAVPO is not mandatory. It is a Holding pattern in lieu of procedure turns. JAVPO is the Intermediate Fix for the approach. You can use JAVPO or its associated intermediate leg as a gate for a straight-in approach on the IAP. Just state your intentions and get clearance to maintain spacing and separation.
Wbeard52 has given a great acronym to remember when a procedure turn is not required. Another one to remember is the one derived from the FAA‘s list of situations not requiring a procedure turn included in the Instrument Flying Handbook. It is the acronym taught to prepare for the FAA written knowledge test and the practical checkride.
S.H.A.R.P.T.T.

Straight in approaches
Hold in lieu of procedure turns
Arc approaches
Radar vectoring
No Pt noted on the approach
Teardrop course reversal approaches
Timed (start) approaches

One point to note is that a procedure turn should be planned anytime the angle inbound to join the approach will result in a turn of greater than 90°. This means that flying from PMM on the 033 radial direct to JAVPO without vectoring would result in a course reversal. However, you could hold on the published hold at PMM to get set up for the approach. This would necessitate holding at a traditionally very busy (aircraft dense) fix, though. Probably something both you and ATC want to avoid for a GA aircraft if possible.

I would plan to cross PMM from the west and then proceed outbound on the 033° to JAVPO and perform the hold in lieu of procedure turn and then proceed inbound.  ATC does not want you to fly almost 180 turn at PMM so coming from the northeast is no good and so it is marked NA (not approved).
A procedure is always required when it is charted with four exceptions.  Think the acronym SNoRT.

S - Straight - In.  ATC clears you using the words "Straight - In."  N12345 proceed direct JAVPO cleared straight-in ILS RWY 26.
No - NoPT on shown on a route on the instrument plate
R - Radar Vectors to final.  N12345 5 miles from JAVPO fly heading 240° till established on the localizer.  Cleared ILS RWY 26 approach.
T - Timed approaches are in use.  Either ATIS will tell you timed approaches are in use or the pilot hears a time to cross the FAF and that would be a good indication that timed approaches are in use.  (Very uncommon)

This is simply saying you can't approach the VOR from the northeast and then do a 180 back to the approach.  Even if you could do that, you still wouldn't want to do that: why fly 22 miles past the 'IF' and then turn around?  That would just be silly.
Ideally, you would either 1) simply enter and fly the hold as your entry method, 2) get "cleared straight-in" from ATC when they issue your clearance, or 3) get radar vectors to the final approach course from ATC.


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How did jet engine aircraft perform compared to the best reciprocating engine aircraft during WW2?
How did the performance of jets in World War II compare to the cutting edge propeller planes (e.g. P-51, Zero's, etc) of the day?

Much longer take-off distance and less endurance, but higher speed at all altitudes with jets. The performance advantage of jets widened with increasing altitude and resulted in a higher service ceiling. Both the Me-262 and the Ar-234 would fly at least 100 km/h faster than any propeller airplane, and the Ar-234 could fly reconnaissance missions over France and England in complete impunity. A Me-262 test variant (262 V9) with a low-profile canopy reached 600 MPH in level flight.
This gave jets the possibility to break off combat at will. They could only be engaged with success when they were flying in for a landing with empty tanks or by chance, when the jet pilot was unaware of his opponent. Since the Me-262 was designed to intercept bomber formations, it was not suited for dogfights. The Heinkel He-280, on the other hand, was evaluated in 1943 against a FW-190 and would easily outturn the propeller aircraft, so this was not an engine-related characteristic.
Note that the first kill of a Me-262 was by a de Havilland Mosquito, a very fast propeller aircraft that could outrun almost all other propeller aircraft of its time.

Can you put any numbers on speed? I found numbers for the prop planes but not the jets. One other thing -- you said they could only be engaged with success by chance or under special conditions when landing, but I noticed (a) the movie Red Tails about the Tuskeegee airmen shows them taking out jet squadrons with P-51 Mustangs and (b) based on Wikipedia it looks like it was mostly the Germans who used jets, yet they were soundly defeated by Allied planes (I think -- I realize I could be mixed up so feel free to correct me). Any truth to this?

@Hack-R As I remember, the kill ratio of the ME-262 vs allied aircraft was about 5 to 1 in favor the of the ME-262. The ME-262 was the first operational jet fighter, and depending on how you define operational, might be considered the only fully operational jet fighter of WWII. The British Gloster Meteor became operational just before the end of the war, but they didn't allow it to be flown over German territory for fear of it being shot down.

@Terry Got it. That makes sense. 5-to-1 is a huge ratio, but it's not like they could never be shot down (if so, we'd all be hailing Hitler!). Thanks for the clarification.

Not true @Hack-R, Germany at the time could not produce the 262 in enough numbers to make enough of a difference, and experienced pilots were in short supply. Very few were shot down, most were lost to accidents and mechanical failure.

@GdD: The production numbers exceeded 1400, but fuel supply was sporadic and many aircraft were sitting around waiting for fuel or engines. Lack of fuel was grounding more aircraft than a lack of pilots.

True, the synthetic fuel program wasn't producing enough. Energy was a huge problem in general, with coal miners being pulled off the eastern front to try and keep up with demand.

@GdD Very interesting stuff. So resources were a constraint, of course. Still, I like to think the Tuskeegee airmen (first black fighter pilots) didn't help the jets much. "We knew the German jets were faster than we were. Instead of going directly after them, we went away from them and then turned into their blind spots.”

@Hack-R  Movies, (particularly those made in the USA) are rarely a reliable guide to what actually happened.

@Hack-R: "Red Tails" is an especially bad example of bad Hollywood-ism, being only loosely related to reality in the first place. The Me-262 scene was particularly atrocious... you just don't survive direct hits from a 30mm canon shell. This is what a 30mm hit looks like...

I don't have much in the way of numbers, but, at least from reading the autobiography of Chuck Yeager ("Yeager: An Autobiography" - http://www.amazon.com/Yeager-Autobiography-Chuck/dp/0553256742) that he shot down an Me 262 (one of Germany's jet fighters) by exploiting a weakness of the aircraft (such as their long landing approaches at relatively slow speeds).  He notes that, as @Peter Kampf says, that it was really hard to engage 262s due to their extremely high speed, but also keep in mind that this doesn't make them a good fighter -- especially not against slower aircraft when both only have guns.  If the aircraft was designed to cruise at high speeds, it would be extremely difficult for it to maneuver at low speeds due to the higher wing loading (e.g., maneuvering loads become higher for a lower airspeed and you become structurally limited on how tight your turns can be).  Hence, a slower aircraft could, potentially, fly circles around such a plane.  If it could convince the jet to slow down in the first place...
However, because of their high speed, Yeager notes that they could dive through a bomber formation with relative impunity.  They would not have long to fire on the formation, but it was incredibly difficult to hit the jet fighter.
Anyhow, according to the National Air and Space Museum's website (http://airandspace.si.edu/collections/artifact.cfm?object=nasm_A19600328000), the Me-262 had a top speed of 540 mph (unclear if that is in level flight, but I'd believe so).  Looking at a max. level flight speed versus altitude for a P-51B, it never breaks 450 mph (source: http://www.wwiiaircraftperformance.org/mustang/p-51b-24771-level-blue.jpg).  This performance plot is from an Army performance test in 1943 (source: http://www.wwiiaircraftperformance.org/mustang/P-51B_Official_Performance_Figures_15May1943.jpg).

The speed advantage of the Me262 gave it an advantage in combat over the piston engined aircraft used by the Allies, once German pilots had developed suitable tactics which exploited this advantage while minimising disadvantages such as the engines' slow throttle response, and the need to operate the throttles slowly to avoid engine failures and flameouts. The Germans found that the best approach to flying the Me262 in combat was to leave the throttles alone as far as possible and use climb and dive tactics. When flown in this way the Me262 was effective against both fighters and bomber aircraft. Pilots learned to avoid turning dogfights with the more manoeuverable Allied fighters.
While the explosive and incendiary shells fired by the Rheinmetall MK108 30mm cannon were very destructive, pilots complained that the relatively low 540 m/s muzzle velocity was a handicap in fighter versus fighter combat and also in ground attack work.

One other factor to consider... while the ME262 had a higher top speed, it would tend to lose a lot of that speed in a tight turn. That, combined with its slower acceleration meant that traditional turning/dogfighting tactics were not the optimal for the jet fighters.
The best ME262 pilots tended to use the same basic tactics that US pilots used in the Pacific against the slower but more nimble Japanese fighters: diving attacks at a high speed the opponent couldn't match, then pulling up to trade speed for altitude at a rate the opponent couldn't match, then reposition and repeat.


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Do pilots ever intentionally takeoff or land downwind?
Is there a reason a pilot would takeoff downwind or land downwind? Is it legal to do this?

I have observationaly confirmed that if you paint a 737 in a Southwest livery and the taxi route for opposite direction takeoff is shorter than to the active runway, that plane will 100% of the time takeoff opposite direction with up to its maximum tailwind limitation.

It's perfectly legal, but check the Chart Supplement (or AFD, or instrument approaches) to see if there are special exceptions for that airport.
If it is a non-towered airport and you aren't assigned a runway, you should of course try to use the prevailing runway that is already in use by other traffic in the pattern when it makes sense.
Some airports have obstructions in one direction of their runways that make it necessary (or safer) to always land or depart in a certain direction.
Some runways have a steep incline in one direction, making it preferable to land upslope and depart downslope.  (For example Lukla, Nepal: YouTube)
It's also common for larger airports to have preferred directions for arrival or departures in order to keep the flight volume high, even when the wind prefers other runways.
However, that usually relates to operating with intentional crosswinds at an airport that has parallel runways in a primary direction (allowing them to conduct more simultaneous flight operations), but not in the intersecting direction.

Well, it's not perfectly legal, it is only legal up to the tailwind limit.

... So, in Lukla, when you're on approach, you could have aircraft taking off at you? That would be my worst nightmare as a pilot. Of course, a glance at Lukla's Wikipedia page would imply that Lukla poses many nightmare scenarios for the average bush pilot.

Airports that are busy enough where the volume of departures and arrivals becomes problematic will generally have a control tower that coordinates that. Smaller airports (like Lukla presumably is) are non-towered and the pilots are listening on the radio to know if any other aircraft are currently planning to land or depart. In both cases, a departing aircraft should not enter the actual runway until it knows that no other aircraft is still using it (or imminently about to).

@KeithS: Yes, you could. It is towered though, so the controller should take care of preventing that. There are worse things about Lukla like the fact that once over the threshold there is basically no chance of going around.

Can you give examples of airports with obstructions in one direction? It seems like such a runway would not permit a go around.

@tas KTEX, KASE, are the first two that come to mind, but there are quite a few "one way in & out" airports like that.

@tas: Another famous is Courchevel / LFLJ with its 18% sloping runway.

Yes, aircraft do take-off and land downwind sometimes. During low wind conditions it is sometimes preferable to use a runway in the downwind direction for noise abatement reasons.
During downwind landing the ground speed will be higher and therefore it takes more distance to stop the aircraft. This increases the risk of a runway excursion.
During downwind departure a higher ground speed is needed to take-off and therefore a longer runway will be required. An other effect is that the climb-out angle is lower due to tailwind, lowering the obstacle clearance and increasing the risk of Controlled Flight Into Terrain (CFIT).
Commercial aircraft have limits one the amount of tailwind that is allowed on take-off and landing. For most aircraft it is 10 or 15 knots.
For more information about the safety aspects of tailwind operations see this report by NLR.

Floatplane operators may occasionally choose to takeoff with a light tailwind when to do otherwise would mean bucking a strong river current. I live on Oregon's McKenzie River, and doing this when operating out of Leaburg Lake (actually just a wide spot in the river) many years ago in a J3 on floats with only 85 horsepower made such takeoffs easier.

Landing downwind is often the only option when flying an instrument approach procedure to minimums (the lowest ceiling at which landing is permitted).
Many airports only have approaches to one runway direction. In the event that the approach is required and circling is not possible, a straight-in downwind landing is the pilot's only option.
For example, given an approach to runway 27 with winds of 8 knots at 080:
Circling MDA      | 1100 feet MSL
Straight-in DA    |  700 feet MSL
Field elevation   |  450 feet MSL

If the clouds are below 1100 feet, a pilot would be unable to circle to the "correct" runway  09; they would have to land straight in to 27, which is a downwind landing.

Given the aircraft I personally fly, I will avoid doing so and divert to an alternate if the tailwind is higher than 10 knots or the runway is shorter than 4500 feet. Tailwind landings are inherently riskier and I'd rather take the delay.

Interesting consideration - my instinct would be the same as yours ("I'm just gonna go land somewhere else instead..."), but with all other options exhausted and assuming adequate pavement for the higher groundspeed a downwind landing would indeed be the thing to do in this case!

Fortunately it's extremely rare to have strong wind and low visibility, because strong wind blows the moisture up and away. So only having low visibility procedures for some runways turns out well in practice (autoland is usually also only certified up to 10 kt crosswind).

@JanHudec that's true, but low ceilings are certainly not incompatible with high winds. That's the scenario I was discussing.

In the case of motorless flight (preferrably intentionally, i.e. with a glider) the choice of landing direction for an outlanding is very strongly influenced by the terrain. Of course it is much preferrable to land with headwind, but in uneven terrain this can be impossible. In sloping terrain, it can be better (or compulsory) to land into direction of the steepest rise.
The same thing also can be the case for landing or (reversed) for takeoff on airports with very pronounced runway slopes (mostly in mountaineous terrain, e.g. in Courchevel), or glacier landings.
In some cases, it is simply convenient to land downwind, again esp. for motorless aircraft to reduce the distance to the hangar - conditions permitting, of course.

I fly small Cessna aircraft, but frequently fly out of airports with 10,000 ft. runways.  (A C-172 typically needs around 1,500 ft).
So when its more convenient for me, I can ask the Tower for an "Opposite Direction Departure", meaning that I want to take off with the wind.
It does take me a little more time to get my airspeed up and take off, but so long as I have plenty of runway in front of me, its not a problem.  And it generally means I can take off more directly towards my course, instead of taking off, flying a pattern around the airport, then getting on course.

You mean ask Ground, since they would be clearing you to taxi to the appropriate end of the runway in preparation of takeoff?

@bovine:  At most airports, you're probably right.   But in my particular case, I was flying out of KBFI. The A-9 intersection was closest to my flight school, and had plenty of space in both directions.  So Ground always sent us to A-9, and we didn't have to pick a direction until we called Tower.

Sometimes the sun is setting directly over the end of the runway with the headwind. Definitely a challenge, and it is a wise decision to land in the other direction unless the tailwind isn't too strong. If it's a controlled airport, you have to get a clearance for that of course.

When I flew light airplanes in Alaska there were a few one-way-in-one-way-out strips that I would take the tailwind before departing.  Very common in some parts of the world.

A good answer would be, "Yes, when operational constraints dictate." A pilot needs to consider the risks when deviating from standard operating procedures and practices. There are reasons airplanes take off into the wind, and they are the same reasons birds takeoff into the wind. Consider that birds have been flying a very long time. As far as I am aware it is not illegal to takeoff downwind within the stipulations of the regulations as noted elsewhere. On landing one can declare an emergency if necessary.
I had over 300 landings on the USS Nimitz and only one was downwind. At home, ashore, I remember only a few hairy crosswind approaches, but no significant downwind approaches.
The carrier often augmented, or made its own wind over the deck. The approach speeds of fighter jets are pretty high. The A7-E came in at around 120 to 140 knots, if my memory serves me correctly. The headwind made approaches safer.
During flight operations in the Mediterranean 2 of my comrades had a midair collision. One ejected into the Mediterranean, and the other diverted to an airbase on Crete. Operational considerations saw the boat turned downwind towards the pilot bobbing up and down in the 60 degree water, and this set up an unusual relative wind over the deck. There were both strong downwind and crosswind components.
A pilot without much shipboard experience came in for the break and by the 45 degree position found himself low and well right of center line. The pilot had been too close abeam for the relative wind, and failed to correct during the remainder of the approach. The LSO ended his approach with the call "Wave off, wave off!"  At this point his altitude was below the top of the Island. He went to full power and wrapped the aircraft up into a 45 degree turn to wave off over the deck. Waving off below and to the right of the Island would have been unusual, but it also would have been safe.
I was watching the approach on the platform cameras in the ready room. I thought to myself that this was the textbook scenario for an approach turn stall. Right after that thought his nose sliced down out of the turn. It was quite dramatic. During training at altitude it is difficult to get a good sense of the quickness of the stall. The A7 impacted the water just behind the ship. Parts of the plane and water rained down on the flight deck. He did not eject.
Think through the departure or approach, and adjust procedures to stay safe. If you will be deviating from standard operating procedures and practices, brief that part of the flight. If the decision has to be made airborne: "aviate, navigate, and communicate."

Commercial passenger operations flying under Part 121 will have an FAA approved OpSpec (Operating Specifications) that will specify the maximum tail wind component for landing.  If the wind is above that, the pilot must land the other way or find a different runway, or risk disciplinary action by his company.


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When has a pilot legally accepted an ATC clearance or instruction?
At what point is a pilot considered to have accepted an clearance or instruction? Upon simple acknowledgement (e.g., Roger or Wilco), any readback, correct readback, or something else?

In what context? IFR, takeoff, landing, Class B, LAHSO? And is there an importance to when it is legally accepted? The language I can think of is "obtained" in 91.123, not accepted, although accepted is used in LAHSO.

There is no such thing as "legally accepted". FAA regulations are REGULATIONS, not laws. The issuer will assume the pilot has "accepted" a clearance as soon as any acknowledgement is given.

I would try to not over complicate this issue.  There are three conditions a clearance/instruction can have:

Transmitted/Received
Accepted
Rejected

If ATC provides a clearance/instruction, but has not yet received a response, the clearance/instruction has not been accepted or rejected.
If the pilot acknowledges the clearance/instruction with a read-back or affirmation of understanding, then it is accepted.
If the pilot responds with the term "unable" to any element of the clearance/instruction, then the entire clearance/instruction set is rejected, which may prompt ATC to inquire as to why the clearance/instruction is being rejected or to provide a different clearance/instruction.
Per 91.3 the pilot-in-command reserves the right to act as the final authority as to the operation of the aircraft, but once a clearance/instruction is accepted, it is an indication that the pilot-in-command is agreeing, as the authority of operation, to operate by that clearance/instruction until released from it or forced to deviate due to an emergency or TCAS alert.
One note for safety: to be clear, one cannot simply reject one element of a clearance/instruction without rejecting the entire clearance/instruction.  For example, if ATC issues a turn and a climb, but the pilot's aircraft cannot or may not climb higher for whatever reason, the pilot should not read back the new heading and say, "unable climb," and then turn to the heading and wait for more instructions.  What if the climb is to avoid obstructions that exist on the new heading?

A clearance doesn't need to be accepted. It's a clearance, not an instruction. You are cleared to do something, not required to do so.

A clearance can be refused based on parameters with which it might be issued.  Even instructions aren't required to be accepted and that is covered in another answer. Either one may be declined if necessary.

For example: a LAHSO clearance is a clearance, and by acknowledging it, you have accepted it.  If you can't land and hold short, you should reply to the controller, "unable" rather than replying, "Runway ##, cleared to land and hold short of runway XX."

This isn't a very satisfying answer, but as far as I can tell (after a lot of searching) there's no formal definition.
A lot of discussions about this start from 14 CFR 61.3:

No person may act as pilot in command of a civil aircraft under IFR or
  in weather conditions less than the minimums prescribed for VFR flight
  unless that person holds [an instrument rating and is current]

If that's what you were thinking about, the universal consensus on pilot forums is that by reading back an IFR clearance you're accepting it, and from that moment the flight is operating under IFR. However, there seems to be no explicit regulation or statement from the FAA to confirm it.
In fact, I couldn't find a definition (P/CG, 14 CFR 1.1 etc.) or FAA legal interpretation of what "accepting" a clearance of any kind actually means. The AIM and the ATC orders both refer to "acknowledging" clearances, e.g. in the ATC Orders (section 2-4-3):

pilots may acknowledge clearances, control instructions, or other
  information by using “Wilco,” “Roger,” “Affirmative,” or other words
  or remarks with their aircraft identification.

And the AIM mentions some scenarios where a readback - as opposed to an acknowledgement - is required, e.g. this from section 4-4-7:

Pilots of airborne aircraft should read back those parts of ATC
  clearances and instructions containing altitude assignments, vectors,
  or runway assignments as a means of mutual verification.

But practically speaking, when ATC issues a clearance or instruction there are only three basic options:

The pilot reads back the clearance correctly (or ATC asks again until he gets it right)
The pilot acknowledges without a readback (and ATC accepts that, or asks for a readback)
The pilot says "unable" or asks for a different clearance

In case 1, it seems reasonable that the correct readback is the acceptance, because the pilot didn't decline the clearance or ask for a different one. In case 2, the acknowledgement is the acceptance for the same reason (I can't think of a good example offhand; see this question). In case 3, the pilot is declining the clearance anyway.
Personally, I think there's no formal definition  because it just isn't necessary. It seems obvious that by reading something back or saying "roger" you're agreeing to follow ATC's instructions, and I think it would be very difficult to argue differently.